A population of wolves (y) in a national park triples every 20 years. The function y=15(3^x) where x = number of 20 year periods that have passed can be used to find the population.

Question

A population of wolves (y) in a national park triples every 20 years.  The function y=15(3^x)  where x = number of 20 year periods that have passed can be used to find the population.

What is the y-intercept of this graph and what does it represent in this situation?

How many wolves would be in the national part after 60 years?

Answer 1

This is an example of exponential growth

The y-intercept is 15. The population of wolves was 15 at the start of record keeping.

The formula says the population triples every 20 years. 60 years/20 = 3 periods

y=15(3^3) y= (15)(3)(3)(3) = 405 wolves

Answer 2

The y-intercept by definition is where x=0 (since in order to cross the y-axis, x must cross from the negatives into the positives and pass over 0). Therefore, to answer this question, we must substitute 0 for x.

y=15(3^0).

Let's start by simplifying the exponent. Any number to the power of 0 will equal 1, so 3^0=1.

y=15(1)

Finally, 15(1)=15, making y=15.

So, what does 15 represent? Well, if X is the number of 20-year cycles, and at the y-intercept, x=0, that means that there have not been any 20-year cycles. Therefore, this must be the initial number of wolves in the national park.

Answer to part 1: the y-intercept is 15 and 15 is the initial number of wolves in the park.

Now for part 2: how many wolves will be in the park after 60 years? To start, we must remember that x does not represent the number of years but instead the number of 20-year cycles. So, how many 20-year cycles are there in 60? Well,

60 years/20 year cycles=3 cycles

Now, we just need to plug this into the equation:

y=15(3^3)

Let's start with the exponent again. You can do this with a calculator, but I'm going to explain how to do it without one. 3^3 means 3 multiplied by itself 3 times, so 3(3)(3). And:

3(3)(3)=9(3)=27

y=15(27)

Now, let's multiply 15 and 27, which is 405. Therefore,

y=405 wolves in the park after 60 years.